Arkiv för Matematik

, 28:273 | Cite as

A multi-dimensional renewal theorem for finite Markov chains

  • Thomas Höglund
Article
  • 42 Downloads

Keywords

Compact Subset Central Limit Theorem Stationary Increment Renewal Theory Martin Boundary 

References

  1. 1.
    Berbeee, H. C. P.,Random walks with stationary increments and renewal theory, Math Centre. Tract. 112, Amsterdam, 1979.Google Scholar
  2. 2.
    Cinlar, E., On semi-Markov processes on arbitrary spaces,Proc. Cambridge Philos. Soc. 66, (1969) 381–392.MATHMathSciNetGoogle Scholar
  3. 3.
    Höglund, T., Central limit theorems and statistical inference for finite Markov chains,Z. Wahrsch. Verw. Gebiete 29, (1974) 123–151.MATHCrossRefGoogle Scholar
  4. 4.
    Höglund, T., A multi-dimensional renewal theorem,Bull. Sc. math., 2 e série,112, (1988) 111–138.MATHGoogle Scholar
  5. 5.
    Höglund, T.,The ruin problem for finite Markov chains, Report, 1989.Google Scholar
  6. 6.
    Iosifescu, M., An extension of the renewal theorem,Z. Wahrsch. Verw. Gebiete 23, (1972) 148–152.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Jacod, J., Théorème de renouvellement et classification pour les chaines semimarkoviennes,Ann. Inst. H. Poincaré Sect. B (N.S.),7, (1971) 83–129.MATHMathSciNetGoogle Scholar
  8. 8.
    Janson, S., Renewal theory form-dependent variables,Ann. Probab. 11, (1983) 558–586.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kato, T.,Perturbation theory for linear operators, Springer, Berlin-Heidelberg-New York, 1966.MATHGoogle Scholar
  10. 10.
    Keilson, J., Wishart, D. M. G., A central limit theorem for processes defined on a finite Markov chain,Proc. Cambridge Philos. Soc. 60, (1964), 547–567.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kesten, H., Renewal theory for functional of a Markov chain with general state space,Ann. Probab. 2, (1974) 355–386.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Miller, H. P., A convexity property in the theory of random variables, defined on a finite Markov chain,Ann. Math. Statist. 32 (1961) 1261–1270.Google Scholar
  13. 13.
    Ney, P. andSpitzer, F., The Martin boundary for random walk,Trans. Amer. Math. Soc. 121, (1966) 116–132.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Orey, S., Change of time scale for Markov processes,Trans. Amer. Math. Soc. 99, (1961) 384–397.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pyke, R., Markov renewal processes with finitely many states,Ann. Math. Stat. 32, (1961) 1243–1259.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Runnenburg, J. T.,On the use of Markov processes in one-server waiting time problems and renewal theory, Thesis, University of Amsterdam, 1960.Google Scholar
  17. 17.
    Spitzer, F.,Principles of Random Walk, Van Nostrand, Toronto-New York-London, 1964.MATHGoogle Scholar

Copyright information

© Institut Mittag-Leffler 1990

Authors and Affiliations

  • Thomas Höglund
    • 1
  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

Personalised recommendations